nLab Euler-Lagrange form




Given a smooth bundle EΣE \to \Sigma with Σ\Sigma a smooth manifold of dimension nn, then a horizontal nn-form L\mathbf{L} on the jet bundle Jet(E)Jet(E) is a local Lagrangian. Its de Rham differential has a unique decomposition into a source form E\mathbf{E} and a horizontally exact form (with respect to the variational bicomplex)

dL=Ed Hθ. d \mathbf{L} = \mathbf{E} - d_H \theta \,.

This source form E\mathbf{E} is the Euler-Lagrange form of L\mathbf{L}. It vanishes precisely at those points which are solutions to the Euler-Lagrange equations induced by L\mathbf{L}.

The combination ρL+θ\rho \coloneqq \mathbf{L} + \theta is the corresponding Lepage form.


  • G. J. Zuckerman, Action principles and global geometry , in Mathematical Aspects of String Theory, S. T. Yau (Ed.), World Scientific, Singapore, 1987, pp. 259€284. (pdf)

Last revised on November 8, 2017 at 20:17:59. See the history of this page for a list of all contributions to it.